Full-state control method for the master-slave robot system with flexible joints and time-varying delays

ABSTRACT

A full-state control method for a master-slave robot system with flexible joints and time-varying delays is provided. In a teleoperation system formed by connecting a master robot and a slave robot through network, a proportional damping controller based on a position error and velocities, and a full-state feedback controller based on backstepping are designed for the master robot and the slave robot, respectively. High-dimension uniform accurate differentiators are designed to realize an exact difference to the virtual controllers. Delay-dependent stability criteria are established by constructing Lyapunov functions. Therefore, the criteria for selecting controller parameters are presented such that the global stability of the master-slave robot system with flexible joints and time-varying delays is realized. For the master-slave robot system with flexible joints, the global precise position tracking performance is realized by adopting a full-state feedback controller based on the backstepping method and the high-dimensional uniform accurate differentiators. Moreover, the global asymptotic convergence of the system is guaranteed and the robustness of the system is improved.

TECHNICAL FIELD

This present invention relates to the control field of the flexiblemaster-slave robot system, in particular to a full-state control methodfor a master-slave robot system with flexible joints and time-varyingdelays.

BACKGROUND

The master-slave robot system with flexible joints is a very complexdynamic system with high-nonlinearity, strong-coupling, and time-varyingdelays. To complete some complex and difficult work, flexible jointmanipulator has the advantages of small volume, high flexibility, highdeadweight ratio of load, low energy consumption, and wider workingspace compared with rigid joint manipulators. In most cases, theteleoperation system is a kind of remote operating system to completemore complex operations. It mainly consists of an operator, a masterrobot system, a network information transmission channel, a slave robotsystem, and external working environment. Its workflow can be roughlydescribed as follows: the operator manipulates the master robotdirectly, so the control instructions will be sent to the slave robotthrough the network information transmission channel, then the slaverobot will act on the remote environment according to the instructions.Meanwhile, the relevant information will also be feedback to theoperator timely. Therefore, a closed-loop teleoperation system can beconstituted and the operating tasks can be completed effectively. Forslave robot systems, n-DOF (Degree Of Freedom) robots with flexiblejoints are usually used to execute commands from the master side.

With the development and progress of science and technology, theexploration of human beings has been constantly expanding. Sinceflexible joints are characterised by small size and high flexibility, aflexible master-slave robot system is more widely used. However,vibration of the flexible joint may affect the control accuracy andsystem stability. Therefore, the proposed control method can not onlyenable the robot to complete the work objectives but also eliminate thevibration of the flexible joint to ensure the stability of the system.

For the master-slave robot system with flexible joints, a full-statefeedback control method based on the backstepping technique is proposed.The position error between the master and slave robots with time delaysare used to analyze the global tracking performance. The globallyaccurate position tracking is realized and the robustness of theclosed-loop system is improved. The high-dimension uniform accuratedifferentiators are employed to realize the precise difference to thevirtual controllers and the convergence of the system is improved.

SUMMARY

The purpose of the invention is to provide a full-state control methodfor the master-slave robot system with flexible joints and time-varyingdelays to solve the problem of position tracking and stability existingin the master-slave robot system with flexible joints. To achieve theseabove objectives, the following technical schemes are adopted.

A full-state control method for the master-slave robot system withflexible joints and time-varying delays includes the following steps:

Step 1: connecting a master robot and a slave robot through the networkto form a teleoperation system.

Step 2: measuring system parameters of the master robot and the slaverobot, among which the position and velocity information of joints andmotors are measured in real-time.

Step 3: designing the first virtual controller and the second virtualcontroller, respectively.

The first virtual controller is X^(*) _(s3)=X_(s1)+S_(s)⁻¹(k_(s)(X_(m1)(t−T_(m)(t))−X_(s1))−α_(s)X_(s2)), the second virtualcontroller is

${X_{s4}^{*} = {{\overset{.}{X}}_{s\; 3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}}},$

the subscripts m and s denotes the master robot and the slave robot,respectively, X*_(s3) and X*_(s4) are virtual controllers, {dot over(X)}*_(s3), is the first derivative of the virtual controller X*_(s3),T_(m)(t) and T_(s)(t) are forward time delay (from the master robot tothe slave robot) and feedback time delay (from the slave robot to themaster robot), respectively, α_(s) is a damping coefficient which is apositive constant, k_(m), k_(s)>0 are proportional coefficients, S_(s)⁻¹ and S_(s) ^(T) are the inverse matrix and the transpose matrix of adiagonal positive-definite constant matrix S_(s) which contains thejoint stiffness of the slave robot, respectively, and k₁ is selected tobe a positive constant.

Step 4: designing high-dimension uniform accurate differentiators tocarry out a precise difference to the first virtual controller and thesecond virtual controller.

Step 41:

letting X₁=X*_(s3), X₂={dot over (X)}*_(s3), σ₁=X₁−Y₁, σ₂=X₂−Y₂,

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{1} = X_{2}} \\{{\overset{.}{X}}_{2} = {\overset{¨}{X}}_{s3}^{*}}\end{matrix} \right. & (1) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{1} = {{\lambda_{1}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} + {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + Y_{2}}} \\{{\overset{.}{Y}}_{2} = {\alpha_{1}\frac{\sigma_{1}}{\sigma_{1}}}}\end{matrix} \right. & (2)\end{matrix}$

with the Equations (1) and (2),

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{1} = {{{- \lambda_{1}}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} - {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + \sigma_{2}}} \\{{\overset{.}{\sigma}}_{2} = {{{- \alpha_{1}}\frac{\sigma_{1}}{\sigma_{1}}} + {\overset{¨}{X}}_{s3}^{*}}}\end{matrix} \right. & (3)\end{matrix}$

is derived, where X*_(s3) denotes the virtual controller, {dot over(X)}*_(s3) denotes the first derivative of the virtual controllerX*_(s3), {umlaut over (X)}_(s3) denotes the second derivative of thevirtual controller X*_(s3), Y₁ is an estimate of the virtual controllerX*_(s3), Y₂ is an estimate of {dot over (X)}*_(s3), σ₁ and σ₂ areestimation errors, λ₁, λ₂, α₁>0 are system control gains, P>1 is aconstant, {umlaut over (X)}*_(s3) supposed to be bounded and satisfies∥{umlaut over (X)}*_(s3)∥≤L₃ with a known constant L₃>0, and if theparameters are selected to satisfy the conditions α₁>4L₃, λ₁>√{squareroot over (2α₁)}, the estimation errors σ₁, σ₂ will globally converge tothe origin quickly, thus a precise difference value {dot over (X)}*_(s3)of X*_(s3) is obtained.

Step 42:

letting X₃=X*_(s4), X₄={dot over (X)}*_(s4), σ₃=X₃−Y₃, σ_(z), =X₄−Y₄

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{3} = X_{4}} \\{{\overset{.}{X}}_{3} = {\overset{¨}{X}}_{s\; 4}^{*}}\end{matrix} \right. & (4) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{3} = {{\lambda_{3}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} + {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + Y_{4}}} \\{{\overset{.}{Y}}_{4} = {\alpha_{2}\frac{\sigma_{3}}{\sigma_{3}}}}\end{matrix} \right. & (5)\end{matrix}$

with the Equations (4) and (5),

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{3} = {{{- \lambda_{3}}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} - {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + \sigma_{4}}} \\{{\overset{.}{\sigma}}_{4} = {{{- \alpha_{2}}\frac{\sigma_{3}}{\sigma_{3}}} + {\overset{¨}{X}}_{s\; 4}^{*}}}\end{matrix} \right. & (6)\end{matrix}$

is derived, where X*_(s4) denotes a virtual controller, {dot over(X)}*_(s4) denotes the first derivative of the virtual controllerX*_(s4), {umlaut over (X)}*_(s4) denotes the second derivative of thevirtual controller X*_(s4), Y₃ is an estimate of the virtual controllerX*_(s4), Y₄ is an estimate of {dot over (X)}*_(s4), σ₃ and σ₄ areestimation errors, λ₃, λ₄, α₂>0 are system control gains, P>1 is aconstant, {umlaut over (X)}*_(s4) is supposed to be bounded ∥{umlautover (X)}*_(s4)∥≤L₄ with a known positive constant L₄>0, and ifparameters are selected to satisfy the conditions α₂>4L₄, λ₃>√{squareroot over (2α₂)}, the estimation errors σ₃, σ₄ will globally converge tothe origin quickly, thus a precise difference value {dot over (X)}*_(s4)of X*_(s4) is obtained.

Step 5: designing the controllers of the master-slave robot system withflexible joints by using the backstepping method as follows

${\tau_{m} = {{- {k_{m}\left( {X_{m1} - {X_{s1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {\alpha_{m}X_{m2}}}}{\tau_{s} = {{S_{s}\left( {X_{s3} - X_{s1}} \right)} + {J_{s}\left( {{\overset{.}{X}}_{s4}^{*} - \left( {X_{s3} - X_{s3}^{*}} \right) - {k_{2}\left( {X_{s4} - X_{s4}^{*}} \right)}} \right)}}}$X_(s3)^(*) = X_(s1) + S_(s)⁻¹(k_(s)(X_(m1)(t − T_(m)(t)) − X_(s1)) − α_(s)X_(s2))$X_{s4}^{*} = {{\overset{.}{X}}_{s3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}}$

where τ_(m) and τ_(s) are control torques provided by the controllers,X*_(s3) and X*_(s4) are virtual controllers, {dot over (X)}*_(s3) and{dot over (X)}*_(s4) are the first derivatives of the virtualcontrollers X*_(s3) and X*_(s4), respectively, T_(m)(t) and T_(s)(t) areforward time delay (from the master robot to the slave robot) andbackward time delay (from the slave robot to the master robot),respectively, α_(m) and α_(s) are damping coefficients which arepositive constants, k_(m), k_(s)>0 are proportional coefficients, S_(s)⁻¹ and S_(s) ^(T) are the inverse matrix and the transpose matrix of adiagonal positive-definite constant matrix S_(s) which contains thejoint stiffness of the slave robot, respectively, J_(s) is a diagonalconstant matrix of the moment of actuator interia, and k₁ and k₂ areselected to be positive constants.

Step 6: establishing the delay-dependent system stability criteria byconstructing Lyapunov Equations, providing the criteria for selectingcontroller parameters, and realizing the global stability of themaster-slave robot system with flexible joints and time-varying delays.

When the controller parameters are selected such that the followinginequalities hold,

${{{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$

the joint and motor velocities {dot over (q)}_(m), {dot over (q)}_(s),{dot over (θ)}_(s) and position error q_(m)−q_(s) of the master-slaverobot system with flexible joints are all bounded.

Furthermore, when a force F_(h) exerted by an operator to the masterrobot and a force F_(e) exerted by the external environment to the slaverobot are both zero, the controllers are designed as follows:

$\begin{matrix}{{\tau_{m} = {{- {k_{m}\left( {X_{m1} - {X_{s1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {\alpha_{m}X_{m2}} + {G_{m}\left( X_{m\; 1} \right)}}}{\tau_{s} = {{{S_{s}\left( {X_{s3} - X_{s1}} \right)} + {{J_{s}\left( {{\overset{.}{X}}_{s4}^{*} - \left( {X_{s3} - X_{s3}^{*}} \right) - {k_{2}\left( {X_{s4} - X_{s4}^{*}} \right)}} \right)}X_{s3}^{*}}} = {X_{s1} + {S_{s}^{- 1}\left( {{k_{s}\left( {{X_{m1}\left( {t - {T_{m}(t)}} \right)} - X_{s1}} \right)} - {\alpha_{s}X_{s2}} + {G_{s}\left( X_{s\; 1} \right)}} \right)}}}}{X_{s4}^{*} = {{\overset{.}{X}}_{s3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}}}} & (7)\end{matrix}$

where τ_(m) and τ_(s) are control torques provided by the controllers,X*_(s3) and X*_(s4) are virtual controllers, {dot over (X)}*_(s3) and{dot over (X)}*_(s4) are the first derivatives of the virtualcontrollers X*_(s3) and X*_(s4), respectively, T_(m)(t) and T_(s)(t) areforward time delay (from the master robot to the slave robot) andbackward time delay (from the slave robot to the master robot),respectively, α_(m) and α_(s) are damping coefficients which arepositive constants, k_(m), k_(s)>0 are proportional coefficients, S_(s)⁻¹ and S_(s) ^(T) are the inverse matrix and the transpose matrix of adiagonal positive-definite constant matrix S_(s) which contains thejoint stiffness of the slave robot, respectively, G_(m)(X_(m1)),G_(s)(X_(s1)) are the gravitational torques of the master robot and theslave robot, J_(s) is a diagonal constant matrix of the moment ofactuator inertia, and k₁ and k₂ are selected to be positive constants.

When the controller parameters are selected such that the followinginequalities hold,

${{{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$

it can be guaranteed that the joint and motor velocities {dot over(q)}_(m), {dot over (q)}_(s), {dot over (θ)}_(s) and the position errorq_(m)−q_(s) of the master-slave robot system with flexible joints willconverge to zero asymptotically and that the global master-slave robotsystem with flexible joints will asymptotically become stable, where Iis the identity matrix, Z⁻¹ and P⁻¹ are the inverse matrices of positivedefinite matrices Z and P, respectively, α_(m) and α_(s) are dampingcoefficients which are positive constants, k_(m), k_(s)>0 areproportional coefficients, and it is supposed that the time delayT_(m)(t) and T_(s)(t) are bounded, i.e. there are positive scalars T_(m) and T _(s), such that the inequalities T_(m)(t)≤T _(m), T_(s)(t)≤T_(s) hold.

Preferably, the system parameters of the master robot and the slaverobot include the length and the mass of the robot manipulators of themaster robot and the slave robot, the positions and the velocities ofthe robot joints and motor of the master robot and the slave robot, andthe force exerted by the operator and the force exerted by the externalenvironment measured by using force sensors.

Preferably, the design process of the first virtual controller and thesecond virtual controller is as follows:

As for the first virtual controller, the first Lyapunov Equation isselected as follows,

$\begin{matrix}{\mspace{79mu}{{V_{1} = {V_{11} + V_{12} + V_{13}}}{V_{11} = {{X_{m2}^{T}{M_{m}\left( X_{m1} \right)}X_{m2}} + {\frac{k_{m}}{k_{s}}X_{s2}^{T}{M_{s}\left( X_{s1} \right)}X_{s2}} + {2\left( {{U_{m}\left( X_{m1} \right)} - \beta_{m}} \right)} + {\frac{2k_{m}}{k_{s}}\left( {{U_{s}\left( X_{s1} \right)} - \beta_{s}} \right)} + {2{\int_{0}^{t}{\left( {{{- {X_{m2}^{T}(\sigma)}}{F_{h}(\sigma)}} + {\frac{k_{m}}{k_{s}}{X_{s2}^{T}(\sigma)}{F_{e}(\sigma)}}} \right)d\;\sigma}}}}}\mspace{20mu}{V_{12} = {{{k_{m}\left( {X_{m1} - X_{s1}} \right)}^{T}\left( {X_{m1} - X_{s1}} \right)V_{13}} = {{\int_{- {\overset{\_}{T}}_{m}}^{0}{\int_{t + \theta}^{t}{{X_{m2}^{T}(\xi)}Z{X_{m2}(\xi)}d\xi d\theta}}} + {\int_{- {\overset{\_}{T}}_{s}}^{0}{\int_{t + \theta}^{t}{{X_{s2}^{T}(\xi)}P{X_{s2}(\xi)}d\;\xi\; d\;\theta}}}}}}}} & (8)\end{matrix}$

where the integral terms satisfy ∫₀ ^(t)−X_(m2) ^(T)(σ)F_(h)(σ)dσ≥0, ∫₀^(t)X_(s2) ^(T)(σ)F_(e)(σ)dσ≥0, Z and P are positive definite matrices,it is supposed that the time delay T_(m)(t) and T_(s)(t) are bounded,i.e. there are positive scalars T _(m) and T _(s), such that T_(m)(t)≤T_(m), T_(s)(t)≤T _(s), M_(m)(X_(m1)) and M_(s)(X_(s1)) are thepositive-definite inertia matrices of the master robot and the slaverobot, respectively, U_(m)(X_(m1)) and U_(s)(X_(s1)) are the potentialenergy of the master robot and the slave robot satisfying

${{G_{m}\left( X_{m1} \right)} = \frac{\partial{U_{m}\left( X_{m1} \right)}}{\partial X_{m1}}},{{G_{s}\left( X_{s1} \right)} = \frac{\partial{U_{s}\left( X_{s1} \right)}}{\partial X_{s1}}},$

there are positive scalars β_(m) and β_(s) such thatU_(m)(X_(m1))≥β_(m), U_(s)(X_(s1))≥β_(s), and k_(m), k_(s)>0 are theproportional coefficients.

The time derivative of V₁₁ is given by

$\begin{matrix}\begin{matrix}{{\overset{.}{V}}_{11} = {{{- 2}k_{m}{X_{m2}^{T}\left( {X_{m1} - {X_{s1}\ \left( {t - {T_{s}\ (t)}} \right)}} \right)}} -}} \\{{2\alpha_{m}X_{m2}^{T}X_{m2}} + {2\frac{k_{m}}{k_{s}}X_{s2}^{T}{S_{s}\left( {X_{s3} - X_{s1}} \right)}}} \\{= {{{- 2}k_{m}{X_{m2}^{T}\left( {X_{m1} - {X_{s\; 1}\ \left( {t - {T_{s}\ (t)}} \right)}} \right)}} - {2\alpha_{m}X_{m2}^{T}X_{m2}} +}} \\{2\frac{k_{m}}{k_{s}}{X_{s2}^{T}\left( {{S_{s}\ \left( {X_{s\; 3} - X_{s3}^{*}} \right)} + {S_{s}X_{s3}^{*}} - {S_{s}X_{s\; 1}}} \right)}}\end{matrix} & (9)\end{matrix}$

where S_(s) is a diagonal positive-definite constant matrix whichcontains the joint stiffness of the slave robot, S_(s) ⁻¹ is the inversematrix of S_(s).

With Equation (9), the first virtual controller

X* _(s3) =X _(s1) +S _(s) ⁻¹(k _(s)(X _(m1)(t−T _(m)(t))−X _(s1))−α_(s)X _(s2)) is derived.

As for the second virtual controller, the second Lyapunov equation isselected as follows,

V ₂ =V ₁+½(X _(s3) −X* _(s3))^(T)(X _(s3) −X* _(s3))  (10)

The time derivative of V₂ is given by

{dot over (V)} ₂ ={dot over (V)} ₁+(X _(s3) −X* _(s3))^(T)(X _(s4) −{dotover (X)}* _(s3))={dot over (V)} ₁+(X _(s3) −X* _(s3))^(T)(X _(s4) −X*_(s4) +X* _(s4) −{dot over (X)}* _(s3))  (11)

With Equation (11), the second virtual controller

$X_{s4}^{*} = {{\overset{.}{X}}_{s3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}}$

is derived.

Preferably, a high-dimensional uniform accurate differentiators isdesigned as follows.

Letting X₁=X*_(s3), X₂={dot over (X)}*_(s3), σ₁=X₁−Y₁, σ₂=X₂−Y₂,

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{1} = X_{2}} \\{{\overset{.}{X}}_{2} = {\overset{¨}{X}}_{s3}^{*}}\end{matrix} \right. & (12) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{1} = {{\lambda_{1}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} + {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + Y_{2}}} \\{{\overset{.}{Y}}_{2} = {\alpha_{1}\frac{\sigma_{1}}{\sigma_{1}}}}\end{matrix} \right. & (13)\end{matrix}$

With Equations (12) and (13),

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{1} = {{{- \lambda_{1}}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} - {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + \sigma_{2}}} \\{{\overset{.}{\sigma}}_{2} = {{{- \alpha_{1}}\frac{\sigma_{1}}{\sigma_{1}}} + {\overset{¨}{X}}_{s3}^{*}}}\end{matrix} \right. & (14)\end{matrix}$

is derived, where X*_(s3) denotes the virtual controller, {dot over(X)}*_(s3) denotes the first derivative of the virtual controllerX*_(s3), {umlaut over (X)}*_(s3) denotes the second derivative of thevirtual controller X*_(s3), Y₁ is an estimate of the virtual controllerX*_(s3), Y₂ is an estimate of {dot over (X)}*₃, σ₁ and σ₂ are estimationerrors, λ₁, λ₂, α₁>0 are system control gains, P>1 is a constant,{umlaut over (X)}*_(s3) is supposed to be bounded and satisfies ∥{umlautover (X)}*_(s3)∥≤L₃ with a known constant L₃>0, and if the parametersare selected to satisfy the conditions α₁>4L₃, λ₁>√{square root over(2α₁)}, the estimation errors σ₁, σ₂ will globally converge to theorigin quickly, thus a precise difference value {dot over (X)}*_(s3) ofX*_(s3) is obtained,

Letting X₃=X*_(s4), X₄={dot over (X)}*_(s4), σ₃=X₃−Y₃, σ₄=X₄−Y₄,

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{3} = X_{4}} \\{{\overset{.}{X}}_{4} = {\overset{¨}{X}}_{s\; 4}^{*}}\end{matrix} \right. & (15) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{3} = {{\lambda_{3}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} + {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + Y_{4}}} \\{{\overset{.}{Y}}_{4} = {\alpha_{2}\frac{\sigma_{3}}{\sigma_{3}}}}\end{matrix} \right. & (16)\end{matrix}$

With Equations (15) and (16),

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{3} = {{{- \lambda_{3}}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} - {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + \sigma_{4}}} \\{{\overset{.}{\sigma}}_{4} = {{{- \alpha_{2}}\frac{\sigma_{3}}{\sigma_{3}}} + {\overset{¨}{X}}_{s\; 4}^{*}}}\end{matrix} \right. & (17)\end{matrix}$

is derived, where X*_(s4) denotes a virtual controller, {dot over(X)}*_(s4) is the first derivative of the virtual controller X*_(s4),{umlaut over (X)}*_(s4) is the second derivative of the virtualcontroller X*_(s4), Y₃ is an estimate of the virtual controller X*_(s4),Y₄ is an estimate of {dot over (X)}*_(s4), σ₃ and σ₄ are estimationerrors, λ₃, λ₄, α₂>0 are system control gains, {umlaut over (X)}*_(s4)supposed to be bounded and satisfies ∥{umlaut over (X)}*_(s4)∥≤L₄ with aknown positive is constant L₄>0, and, if the parameters are selected tosatisfy the conditions α₂>4L₄, λ₃>√{square root over (2α₂)}, theestimation errors σ₃, σ₄ will globally converge to the origin quickly,thus a precise difference value {dot over (X)}*_(s4) of X*_(s4) isobtained.

Preferably, a delay-dependent system stability criteria is establishedby constructing the Lyapunov Equations, thus a criteria for selectingcontroller parameters are provided. The detailed steps are as follows.

S1 selecting the first Lyapunov Equation as follows,

$\begin{matrix}{\mspace{79mu}{{V_{1} = {V_{11} + V_{12} + V_{13}}}{V_{11} = {{X_{m2}^{T}{M_{m}\left( X_{m1} \right)}X_{m2}} + {\frac{k_{m}}{k_{s}}X_{s2}^{T}{M_{s}\left( X_{s1} \right)}X_{s2}} + {2\left( {{U_{m}\left( X_{m1} \right)} - \beta_{m}} \right)} + {\frac{2k_{m}}{k_{s}}\left( {{U_{s}\left( X_{s1} \right)} - \beta_{s}} \right)} + {2{\int_{0}^{t}{\left( {{{- {X_{m2}^{T}(\sigma)}}{F_{h}(\sigma)}} + {\frac{k_{m}}{k_{s}}{X_{s2}^{T}(\sigma)}{F_{e}(\sigma)}}} \right)d\;\sigma}}}}}\mspace{20mu}{V_{12} = {{{k_{m}\left( {X_{m1} - X_{s1}} \right)}^{T}\left( {X_{m1} - X_{s1}} \right)V_{13}} = {{\int_{- {\overset{\_}{T}}_{m}}^{0}{\int_{t + \theta}^{t}{{X_{m2}^{T}(\xi)}Z{X_{m2}(\xi)}d\xi d\theta}}} + {\int_{- {\overset{\_}{T}}_{s}}^{0}{\int_{t + \theta}^{t}{{X_{s2}^{T}(\xi)}P{X_{s2}(\xi)}d\;\xi\; d\;\theta}}}}}}}} & (8)\end{matrix}$

where the integral terms satisfy ∫₀ ^(t)−X_(m2) ^(T)(σ)F_(h)(σ)dσ≥0, ∫₀^(t)X_(s2) ^(T)(σ)F_(e)(σ)dσ≥0, Z and P are positive definite matrices,it is supposed that the time delay T_(m)(t) and T_(s)(t) are bounded,i.e. there are positive scalars T _(m) and T _(s), such that T_(m)(t)≤T_(m), T_(s)(t)≤T _(s), M_(m)(X_(m1)) and M_(s)(X_(s1)) are thepositive-definite inertia matrices of the master robot and the slaverobot, respectively, U_(m)(X_(m1)) and U_(s)(X_(s1)) are the potentialenergy of the master robot and the slave robot satisfying

${{G_{m}\left( X_{m1} \right)} = \frac{\partial{U_{m}\left( X_{m1} \right)}}{\partial X_{m1}}},{{G_{s}\left( X_{s1} \right)} = \frac{\partial{U_{s}\left( X_{s1} \right)}}{\partial X_{s1}}},$

there are positive scalars β_(m) and β_(s) such thatU_(m)(X_(m1))≤β_(m), U_(s)(X_(s1))≥β_(s), and k_(m), k_(s)>0 areproportional coefficients.

The time derivative of V₁ is given by

$\begin{matrix}{{\overset{.}{V}}_{1} = {{{\overset{.}{V}}_{11} + {\overset{.}{V}}_{12} + {\overset{.}{V}}_{13}} \leq {{{X_{m2}^{T}\left( {{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m2}} + {{X_{s2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s2}} + {2\frac{k_{m}}{k_{s}}X_{s2}^{T}{S_{s}\left( {X_{s3} - X_{s3}^{*}} \right)}}}}} & (19)\end{matrix}$

where I is the identity matrix, Z⁻¹ and P⁻¹ are the inverse matrices ofpositive definite matrices Z and P, respectively, S_(s) is a diagonalpositive-definite constant matrix which contains the joint stiffness ofthe slave robot, α_(m) and α_(s) are damping coefficients which arepositive constants, k_(m), k_(s)>0 are proportional coefficients, it issupposed that the time delay T_(m)(t) and T_(s)(t) are bounded, i.e.there are positive scalars T _(m) and T _(s), such that T_(m)(t)≤T _(m),T_(s)(t)≤T _(s), and X*_(s3) is the first virtual controller.

S2 selecting the second Lyapunov Equation as follows:

V ₂ =V ₁+½(X _(s3) −X* _(s3))^(T)(X _(s3) −X* _(s3))  (20)

The time derivative of V₂ is given by

$\begin{matrix}{{\overset{.}{V}}_{2} \leq {{{X_{m2}^{T}\left( {{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m2}} + {{X_{s2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s2}} + {\left( {X_{s3} - X_{s3}^{*}} \right)^{T}\left( {X_{s4} - X_{s\; 4}^{*}} \right)} - {{k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}^{T}\left( {X_{s3} - X_{s3}^{*}} \right)}}} & (21)\end{matrix}$

S3 selecting the third Lyapunov Equation as follows:

V ₃ =V ₂+½(X _(s4) −X* _(s4))^(T)(X _(s4) −X* _(s4))  (22)

The time derivative of V₃ is given by

$\begin{matrix}{{\overset{.}{V}}_{3} \leq {{{X_{m2}^{T}\left( {{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m2}} + {{X_{s2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s2}} - {{k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}^{T}\left( {X_{s3} - X_{s3}^{*}} \right)} - {{k_{2}\left( {X_{s4} - X_{s\; 4}^{*}} \right)}^{T}\left( {X_{s4} - X_{s\; 4}^{*}} \right)}}} & (23)\end{matrix}$

When the controller parameters are selected such that the followinginequalities hold,

${{{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {\overset{¯}{T_{s}}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {\overset{¯}{T_{s}}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$

the joint and motor velocities {dot over (q)}_(m), {dot over (q)}_(s),{dot over (θ)}_(s) and the position error q_(m)−q_(s) of themaster-slave robot system with flexible joints are all bounded.

When a force F_(h) exerted by an operator to the master robot and aforce F_(e) exerted by the external environment to the slave robot areboth zero, the controllers are designed as follows:

$\begin{matrix}{{\tau_{m} = {{- {k_{m}\left( {X_{m1} - {X_{s1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {\alpha_{m}X_{m2}} + {G_{m}\left( X_{m1} \right)}}}{\tau_{s} = {{S_{s}\left( {X_{s3} - X_{s1}} \right)} + {J_{s}\left( {{\overset{.}{X}}_{s\; 4}^{*} - \left( {X_{s3} - X_{s3}^{*}} \right) - {k_{2}\left( {X_{s4} - X_{s4}^{*}} \right)}} \right)}}}{X_{s\; 3}^{*} = {X_{s1} + {S_{s}^{- 1}\left( {{k_{s}\left( {{X_{m1}\left( {t - {T_{m}(t)}} \right)} - X_{s1}} \right)} - {\alpha_{s}X_{s2}} + {G_{s}\left( X_{s1} \right)}} \right)}}}X_{s\; 4}^{*} = {{{\overset{.}{X}}^{*}}_{s3} - {2\frac{k_{m}}{k_{s}}S_{S}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - {X^{*}}_{s3}} \right)}}} & (24)\end{matrix}$

where τ_(m) and τ_(s) are control torques provided by the controllers,X*_(s3) and X*_(s4) are virtual controllers, {dot over (X)}*_(s3) and{dot over (X)}*_(s4) are the first derivatives of the virtualcontrollers X*_(s3) and X*_(s4), respectively, T_(m)(t) and T_(s)(t) areforward time delay (from the master robot to the slave robot) andfeedback time delay (from the slave robot to the master robot),respectively, α_(m) and α_(s) are damping coefficients which arepositive constants, k_(m), k_(s)>0 are proportional coefficients, S_(s)⁻¹ and S_(s) ^(T) are the inverse matrix and the transpose matrix of adiagonal positive-definite constant matrix S_(s) which contains thejoint stiffness of the slave robot, respectively, G_(m)(X_(m1)),G_(s)(X_(s1)) are the gravitational torques of the master robot and theslave robot, respectively, and k₁ and k₂ are selected to be positiveconstants.

When the controller parameters α_(m), α_(s), k_(m), k_(s), I, T _(m), T_(s), Z, P are selected such that the following conditions hold,

${{{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {\overset{¯}{T_{s}}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {\overset{¯}{T_{s}}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$

it is guaranteed that the joint and motor velocities {dot over (q)}_(m),{dot over (q)}_(s), {dot over (θ)}_(s) and the position errorq_(m)−q_(s) of the master-slave robot system with flexible joints willconverge to zero asymptotically and that the global master-slave robotsystem with flexible joints will asymptotically become stable.

Compared with the existing methods, the present invention has thefollowing advantages:

1. Compared with the local state control method, the proposed full-statecontrol method can realize the global asymptotic stability of themaster-slave system better.

2. The exact position tracking accuracy can be guaranteed by applyingthe full-state feedback controller based on the backstepping methodglobally and the robustness of closed-loop system also can be improved.

3. The precise difference to the virtual controllers can be realized bydesigning high-dimension uniform accurate differentiators. Additionally,this method is more applicable to the high-order control systems.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic control block diagram of the present invention.

DETAILED DESCRIPTION

The invention is further explained in combination with the attachedFIGURE as follows.

As shown in FIG. 1, the steps of the invention method are as follows:

Step 1: connecting the master robot and the slave robot through thenetwork to form a teleoperation system, measuring the system parametersof the master robot and the slave robot, and measuring a force exertedby an operator and a force exerted by the external environment by usingforce sensors.

The system parameters of the master robot and the slave robot includesthe length and the mass of the manipulators, the positions and thepositive-definite inertia matrices M_(m)(X_(m1)) and M_(s)(X_(s1)), thematrices of centripetal and coriolis torques C_(m)(q_(m), {dot over(q)}_(m)) and C_(s)(q_(s), {dot over (q)}_(s)), the gravity torquesG_(m)(X_(m1)) and G_(s)(X_(s1)), the diagonal constant matrix J_(s) ofthe moment of actuator inertia of the master robot and the slave robot,and a diagonal positive-definite constant matrix S_(s) that contains thejoint stiffness of the slave robot respectively according to the lengthand the mass of the manipulators. Moreover, a force F_(h) exerted by theoperator to the master robot and a force F_(e) exerted by the externalenvironment to the slave robot are measured by using force sensors.

The system dynamics equation can be described as

M _(m)(q _(m)){umlaut over (q)} _(m) +C _(m)(q _(m) ,{dot over (q)}_(m)){dot over (q)} _(m) +G _(m)(q _(m))=τ_(m) +F _(h)

M _(s)(q _(s)){umlaut over (q)} _(s) +C _(s)(q _(s) ,{dot over (q)}_(s)){dot over (q)} _(s) +G _(s)(q _(s))=S _(s)(θ_(s) −q _(s))−F _(e)

J _(s){umlaut over (θ)}_(s) +S _(s)(θ_(s) −q _(s))=τ_(s)  (1)

where the subscripts m and s denotes the master robot and the slaverobot, respectively. q_(m), q_(s)ϵR^(n) are the vectors of jointdisplacements. {dot over (q)}_(m), {dot over (q)}_(s)ϵR^(n) are thevectors of joint velocities. {umlaut over (q)}_(m), {umlaut over(q)}_(s)ϵR^(n) are the vectors of joint accelerations. θ_(s)ϵR^(n) isthe vector of motor displacements, {umlaut over (θ)}_(s)ϵR^(n) is thevector of motor accelerations. M_(m)(q_(m)), M_(s)(q_(s))ϵR^(n×n) arethe positive-definite inertia matrices of the system, C_(m)(q_(m), {dotover (q)}_(m)), C_(s)(q_(s), {umlaut over (q)}_(s))ϵR^(n) are thematrices of centripetal and coriolis torques, G_(m)(q_(m)),G_(s)(q_(s))ϵR^(n) are the gravitational torques. J_(s)ϵR^(n×n) is thediagonal constant matrix of the moments of actuator inertia. S_(s)ϵR^(n)is a diagonal positive-definite constant matrix that contains the jointstiffness of the slave robot. F_(h), F_(e)ϵR^(n) are the force exertedby the operator to the master robot and the force exerted by theexternal environment to the slave robot, respectively. τ_(m),τ_(s)ϵR^(n) are control torques provided by the controllers.

Step 2: measuring the position and velocity information of joints andmotors in real-time, designing a proportional damping controller for themaster robot based on position error and velocities, and designing afull-state feedback controller for the slave robot based on backsteppingrecursive technology in combination with Lyapunov equation.

For the master robot with rigid joints, letting X_(m1)=q_(m),X_(m2)={dot over (q)}_(m), the state space expression of the system isobtained as follows:

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{m1} = X_{m2}} \\{{\overset{.}{X}}_{m2} = {{M_{m}^{- 1}\left( X_{m1} \right)}\left( {\tau_{m} + F_{h} - {{C_{m}\left( {X_{m1},X_{m2}} \right)}X_{m2}} - {G_{m}\left( X_{m1} \right)}} \right)}}\end{matrix} \right. & (2)\end{matrix}$

where the subscripts m denotes the master robot, q_(m)ϵR^(n) is thevector of joint displacements, {dot over (q)}_(m)ϵR^(n) is the vector ofjoint velocities, M_(m) ⁻¹(X_(m1)) is the inverse matrix of thepositive-definite inertia matrix M_(m)(X_(m1)), τ_(m)ϵR^(n) is a controltorque provided by the controller, C_(m)(q_(m), {dot over(q)}_(m))ϵR^(n) is the matrix of centripetal and coriolis torque,G_(m)(q_(m))ϵR^(n) is the gravitational torque, and F_(h)ϵR^(n) is aforce exerted by an operator to the master robot.

For the slave robot with flexible joints, letting X_(s1)=q_(s),X_(s2)={dot over (q)}_(s), X_(s3)=θ_(s), X_(s4)={dot over (θ)}_(s), thestate space expression of the system is obtained as follows:

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{s1} = X_{s2}} \\{{\overset{.}{X}}_{s2} = {{M_{s}^{- 1}\left( X_{s1} \right)}\left( {{S_{s}\left( {X_{s3} - X_{s1}} \right)} - F_{e} - {{C_{s}\left( {X_{s1},X_{s2}} \right)}X_{s2}} - {G_{s}\left( X_{s1} \right)}} \right)}} \\{{\overset{.}{X}}_{s3} = X_{s4}} \\{{\overset{.}{X}}_{s4} = {J_{s}^{- 1}\left( {\tau_{s} - {S_{s}\left( {X_{s3} - X_{s1}} \right)}} \right)}}\end{matrix} \right. & (3)\end{matrix}$

where the subscripts s denotes the master robot, q_(s)ϵR^(n) is thevector of joint displacements, {dot over (q)}_(s)ϵR^(n) is the vector ofjoint velocities, θ_(s)ϵR^(n) is the vector of motor displacements, {dotover (θ)}_(s)ϵR^(n) is the vector of motor velocities. M_(s) ⁻¹(X_(s1))is the inverse matrix of the positive-definite inertia matrixM_(s)(X_(s1)), J_(s) ⁻¹ is the inverse matrix of the diagonal constantmatrix J_(s) of the moments of actuator interia, τ_(s)ϵR^(n) is thecontrol torque of the controller, C_(s)(q_(s), {dot over (q)}_(s))ϵR^(n)is the matrix of centripetal and coriolis torque, G_(s)(q_(s))ϵR^(n) isthe gravitational torque, F_(e) is a force exerted by the externalenvironment to the slave robot, and, S_(s)ϵR^(n×n) is a diagonalpositive-definite constant matrix that contains the joint stiffness ofthe slave robot.

The first Lyapunov Equation is selected as follows,

$\begin{matrix}{\mspace{79mu}{{V_{1} = {V_{11} + V_{12} + V_{13}}}{V_{11} = {{X_{m2}^{T}{M_{m}\left( X_{m1} \right)}X_{m2}} + {\frac{k_{m}}{k_{s}}X_{s2}^{T}{M_{S}\left( X_{s1} \right)}X_{s2}} + {2\left( {{U_{m}\left( X_{m1} \right)} - \beta_{m}} \right)} + {\frac{2k_{m}}{k_{s}}\left( {{U_{s}\left( X_{s1} \right)} - \beta_{s}} \right)} + {2{\int_{0}^{t}{\left( {{{- {X_{m2}^{T}(\sigma)}}{F_{h}(\sigma)}} + {\frac{k_{m}}{k_{s}}{X_{s2}^{T}(\sigma)}{F_{e}(\sigma)}}} \right)d\;\sigma}}}}}\mspace{79mu}{V_{12} = {{k_{m}\left( {X_{m1} - X_{s1}} \right)}^{T}\left( {X_{m1} - X_{s1}} \right)}}{V_{13} = {{\int_{- {\overset{¯}{T}}_{m}}^{0}{\int_{t + \theta}^{t}{{X_{m2}^{T}(\xi)}{{PX}_{m2}(\xi)}d\xi d\theta}}} + {\int_{- \overset{¯}{T_{s}}}^{0}{\int_{t + \theta}^{t}{{X_{s2}^{T}(\xi)}P{X_{s2}(\xi)}d\;\xi\; d\;\theta}}}}}}} & (4)\end{matrix}$

where the integral terms satisfy ∫₀ ^(t)−X_(m2) ^(T)(σ)F_(h)(σ)dσ≥0, ∫₀^(t)X_(s2) ^(T)(σ)F_(e)(σ)dσ≥0. Z and P are positive definite matrices.It is supposed that the time delay T_(m)(t) and T_(s)(t) are bounded,i.e. there are positive scalars T _(m) and T _(s), such that T_(m)(t)≤T_(m), T_(s)(t)≤T _(s). M_(m)(M_(m1)) and M_(s)(X_(s1)) arepositive-definite inertia matrices of the master robot and the slaverobot, respectively.U_(m)(X_(m1)) and U_(s)(X_(s1)) are the potential energy of the masterrobot and the slave robot satisfying

${{G_{m}\left( X_{m1} \right)} = \frac{\partial{U_{m}\left( X_{m1} \right)}}{\partial X_{m1}}},{{G_{s}\left( X_{s1} \right)} = {\frac{\partial{U_{s}\left( X_{s1} \right)}}{\partial X_{s1}}.}}$

There are positive scalars β_(m) and β_(s) such thatU_(m)(X_(m1))≥β_(m), U_(s)(X_(s1))≥β_(s), and k_(m), k_(s)>0 areproportional coefficients.

The time derivatives of V₁, V₁₁, V₁₂, V₁₃ are given by

$\begin{matrix}\begin{matrix}{{\overset{.}{V}}_{11} = {2{X_{m2}^{T}\left( {F_{k} - {k_{m}\left( {X_{m1} - {X_{s\; 1}\left( {t - {T_{s}(t)}} \right)}} \right)} - {\alpha_{m}X_{m\; 2}} - {{C_{m}\left( {X_{m1},X_{m2}} \right)}X_{m2}} - {G_{m}\left( X_{m\; 1} \right)}} \right)}}} \\{{{+ 2}\frac{k_{m}}{k_{s}}{X_{s2}^{T}\left( {s,{\left( {X_{s3} - X_{s1}} \right) - \ F_{e}\  - {{C_{s}\left( {X_{s1},X_{2}} \right)}X_{2}} - {G_{s}\left( X_{s1} \right)}}} \right)}} + {\frac{2k_{m}}{k_{s}}{G_{s}\left( X_{s1} \right)}X_{s\; 2}^{T}}} \\{{{+ X_{m2}^{T}}{{\overset{.}{M}}_{m}\left( X_{m1} \right)}X_{m2}} + {\frac{k_{m}}{k_{s}}X_{s2}^{T}{{\overset{.}{M}}_{s}\left( X_{s\; 1} \right)}X_{s2}} + {2{G_{m}\left( X_{m1} \right)}X_{m2}^{T}} - {2X_{m2}^{T}F_{k}} + {2\frac{k_{m}}{k_{s}}X_{s2}^{T}F_{e}}} \\{= {{{- 2}k_{m}{X_{m2}^{T}\left( {X_{m1} - {X_{1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {2\alpha_{m}X_{m2}^{T}X_{m2}} + {2\frac{k_{m}}{k_{s}}X_{s2}^{T}{S_{s}\left( {X_{s3} - x_{s1}} \right)}}}} \\{= {{{- 2}k_{m}{X_{m2}^{T}\left( {X_{m1} - {X_{s\; 1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {2\alpha_{m}X_{m2}^{T}X_{m2}} + {2\frac{k_{m}}{k_{s}}{X_{s2}^{T}\left( {{S_{s}\left( {X_{s\; 3}\  - X_{s\; 3}^{*}} \right)} + {S_{s}X_{s\; 3}^{*}} - {S_{s}X_{s\; 1}}} \right)}}}}\end{matrix} & (5)\end{matrix}$

With Equation (5), the first virtual controller is derived asX*_(s3)=X_(s1)±S_(s) ⁻¹(k_(s)(X_(m1)(t−T_(m)(t))−X_(s1))−α_(s)X_(s2)).By substituting the first virtual controller X*_(s3) into Equation (5),

$\begin{matrix}{\begin{matrix}{{\overset{.}{V}}_{11} = {{{- 2}k_{m}{X_{m2}^{T}\left( {X_{m1} - {X_{s1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {2\alpha_{m}X_{m2}^{T}X_{m2}}}} \\{{+ 2}\frac{k_{m}}{k_{s}}{X_{s\; 2}^{T}\left( {{S_{s}\left( {X_{s3} - X_{s\; 3}^{*}} \right)} + {k_{s}\left( {{X_{m1}\left( {t - {T_{m}(t)}} \right)} - X_{s\; 1}} \right)} - {\alpha_{s}X_{s2}}} \right)}} \\{= {{{- 2}\alpha_{m}X_{m2}^{T}X_{m2}} - {2\frac{k_{m}\alpha_{s}}{k_{s}}X_{s2}^{T}X_{2}} - {2k_{m}{X_{m2}^{T}\left( {X_{m1} - X_{1}} \right)}} - {2k_{m}{X_{m2}^{T}\left( {X_{s1} - {X_{1}\left( {t - {T_{s}(t)}} \right)}} \right)}}}} \\{{{- 2}k_{m}{X_{s2}^{T}\left( {X_{s1} - X_{m1}} \right)}} - {2k_{m}{X_{s2}^{T}\left( {X_{m1} - {X_{m1}\left( {t - {T_{m}(t)}} \right)}} \right)}} + {2\frac{k_{m}}{k_{s}}X_{2}^{T}{S_{s}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}}} \\{= {{{- 2}\alpha_{m}X_{m2}^{T}X_{m2}} - {2{k_{m}\left( {X_{m1} - X_{1}} \right)}^{T}\left( {X_{m2} - X_{s\; 2}} \right)} - {2\frac{k_{m}\alpha_{s}}{k_{s}}X_{s2}^{T}X_{s2}} - {2k_{m}X_{m2}^{T}{\int_{t - {T_{s}{(t)}}}^{t}{{X_{s\; 2}(\xi)}d\;\xi}}}}} \\{{{- 2}k_{m}X_{s2}^{T}{\int_{t - {T_{m}{(t)}}}^{t}{{X_{m2}(\xi)}d\;\xi}}} + {2\frac{k_{m}}{k_{s}}X_{2}^{T}{S_{S}\left( {X_{s3} - X_{s\; 3}^{*}} \right)}}}\end{matrix}\mspace{79mu}{{\overset{.}{V}}_{12} = {2{k_{m}\left( {X_{m2} - X_{s2}} \right)}^{T}\left( {X_{m1} - X_{s1}} \right)}}\begin{matrix}{{\overset{.}{V}}_{13} = {{{\overset{\_}{T}}_{m}X_{m2}^{T}ZX_{m2}} - {\int_{t - {\overset{¯}{T}}_{m}}^{t}{{X_{m2}^{T}(\xi)}{{ZX}_{m2}(\xi)}d\xi}} + {\overset{¯}{T_{s}}X_{s2}^{T}PX_{s2}} - {\int_{t - {\overset{¯}{T}}_{s}}^{t}{{X_{s2}^{T}(\xi)}P{X_{s2}(\xi)}d\;\xi}}}} \\{\leq {{{\overset{\_}{T}}_{m}X_{m2}^{T}{ZX}_{m2}} - {\int_{t - {T_{m}{(t)}}}^{t}{{X_{m2}^{T}(\xi)}{{ZX}_{m2}(\xi)}d\xi}} + {\overset{¯}{T_{s}}X_{2}^{T}PX_{s\; 2}} - {\int_{t - {T_{m}{(t)}}}^{t}{{X_{s2}^{T}(\xi)}P{X_{s2}(\xi)}d\;\xi}}}}\end{matrix}} & (6)\end{matrix}$

is obtained. Furthermore, with the inequality, it holds that

$\begin{matrix}{{{{{- 2}k_{m}X_{m2}^{T}{\int_{t - {T_{s}{(t)}}}^{t}{{X_{s2}(\xi)}d\;\xi}}} - {\int_{t - {T_{s}{(t)}}}^{t}{{X_{s2}^{T}(\xi)}P{X_{s2}(\xi)}d\xi}}} \leq {{\overset{¯}{T_{s}}k_{m}^{2}{X_{m2}^{T}(t)}P^{- 1}{X_{m2}(t)}} - {2k_{m}X_{s2}^{T}{\int_{t - {T_{s}{(t)}}}^{t}{{X_{m2}(\xi)}d\;\xi}}} - {\int_{t - {T_{s}{(t)}}}^{t}{{X_{m2}^{T}(\xi)}{{ZX}_{m2}(\xi)}d\;\xi}}} \leq {{\overset{\_}{T}}_{m}k_{m}^{2}{X_{s2}^{T}(t)}Z^{- 1}{X_{s2}(t)}}}{So}} & (7) \\\begin{matrix}{{\overset{.}{V}}_{1} = {{{\overset{.}{V}}_{11} + {\overset{.}{V}}_{12} + {\overset{.}{V}}_{13}} \leq {{X_{m2}^{T}\left( {{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {\overset{¯}{T_{s}}k_{m}^{2}P^{- 1}}} \right)}X_{m2}}}} \\{{{+ {X_{s2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {\overset{¯}{T_{s}}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}}X_{s2}} + {2\frac{k_{m}}{k_{s}}X_{s2}^{T}{S_{S}\left( {X_{s3} - X_{s\; 3}^{*}} \right)}}}\end{matrix} & (8)\end{matrix}$

is derived, where I is the identity matrix, z⁻¹ and P⁻¹ are the inversematrices of positive definite matrices Z and P, respectively. S_(s) is adiagonal positive-definite constant matrix which contains the jointstiffness of the slave robot. α_(m) and α_(s) are damping coefficientswhich are positive constants. k_(m), k_(s)>0 are proportionalcoefficients. It is supposed that the time delay T_(m)(t) and T_(s)(t)are bounded, i.e. there are positive scalars T _(m) and T _(s), suchthat T_(m)(t)≤T _(m), T_(s)(t)≤T _(s), X*_(s3) is the first virtualcontroller.

The second Lyapunov Equation is selected as follows,

V ₂ =V ₁+½(X _(s3) −X* _(s3))^(T)(X _(s3) −X* _(s3))  (9)

The time derivative of V₂ is given by

{dot over (V)} ₂ ={dot over (V)} ₁+(X _(s3) −X* _(s3))^(T)(X _(s4) −{dotover (X)}* _(s3))={dot over (V)} ₁+(X _(s3) −X* _(s3))^(T)(X _(s4) −X*_(s4) +X* _(s4) −{dot over (X)}* _(s3))  (10)

With equation (10), the second virtual controller is derived as

$X_{s\; 4}^{*} = {{\overset{.}{X}}_{s\; 3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {{k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}.}}$

By substituting the second virtual controller X*_(s4) into Equation(10),

$\begin{matrix}{{\overset{.}{V}}_{2} \leq {{{X_{m2}^{T}\left( {{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m2}} + {{X_{s2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {\overset{¯}{T_{s}}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s2}} + {\left( {X_{s3} - X_{s\; 3}^{*}} \right)^{T}\left( {X_{s4} - X_{s\; 4}^{*}} \right)} - {{k_{1}\left( {X_{s3} - X_{s\; 3}^{*}} \right)}^{T}\left( {X_{s3} - X_{s\; 3}^{*}} \right)}}} & (11)\end{matrix}$

is derived.

The third Lyapunov Equation is selected as follows,

V ₃ =V ₂+½(X _(s4) −X* _(s4))^(T)(X _(s4) −X* _(s4))  (12)

The time derivative of V₃ is given by

$\begin{matrix}\begin{matrix}{{\overset{.}{V}}_{3} = {{\overset{.}{V}}_{2} + {\left( {X_{s4} - X_{s\; 4}^{*}} \right)^{T}\left( {{\overset{.}{X}}_{s4} - {\overset{.}{X}}_{s4}} \right)}}} \\{= {{\overset{.}{V}}_{2} + {\left( {X_{s4} - X_{s\; 4}^{*}} \right)^{T}\left( {{J_{s}^{- 1}\left( {\tau_{s} - {S_{s}\left( {X_{s3} - X_{s1}} \right)}} \right)} - X_{s\; 4}^{*}} \right)}}}\end{matrix} & (13)\end{matrix}$

The full-state feedback controller is derived from Equation (13) as

τ_(s)=S_(s)(X_(s3)−X_(s1))+J_(s)({dot over(X)}*_(s4)−(X_(s3)−X*_(s3))−k₂(X_(s4)−X*_(s4))). By substituting theobtained all-state feedback controller τ_(s) into Equation (13),

$\begin{matrix}{{\overset{.}{V}}_{3} \leq {{{X_{m2}^{T}\left( {{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m2}} + {{X_{s2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {\overset{¯}{T_{s}}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s2}} - {{k_{1}\left( {X_{s3} - X_{s\; 3}^{*}} \right)}^{T}\left( {X_{s3} - X_{s\; 3}^{*}} \right)} - {{k_{2}\left( {X_{s4} - X_{s\; 4}^{*}} \right)}^{T}\left( {X_{s4} - X_{s\; 4}^{*}} \right)}}} & (14)\end{matrix}$

is derived.

The controllers of the master-slave robot system with flexible jointsare obtained by using the backstepping method as

$\begin{matrix}{{\tau_{m} = {{- {k_{m}\left( {X_{m1} - {X_{s1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {\alpha_{m}X_{m2}}}}{\tau_{s} = {{S_{s}\left( {X_{s3} - X_{s1}} \right)} + {J_{s}\left( {{\overset{.}{X}}_{s4}^{*} - \left( {X_{s3} - X_{s3}^{*}} \right) - {k_{2}\left( {X_{s4} - X_{s4}^{*}} \right)}} \right)}}}{X_{s3}^{*} = {X_{s1} + {S_{s}^{- 1}\left( {{k_{s}\left( {{X_{m1}\left( {t - {T_{m}(t)}} \right)} - X_{s1}} \right)} - {\alpha_{s}X_{s2}}} \right)}}}{X_{s\; 4}^{*} = {{\overset{.}{X}}_{s\; 3}^{*} - {2\frac{k_{m}}{k_{s}}S_{S}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s\; 3}^{*}} \right)}}}} & (15)\end{matrix}$

where τ_(m) and τ_(s) are control torques provided by the controllers,X*_(s3) and X*_(s4) are the first virtual controller and the secondvirtual controller, respectively, {dot over (X)}*_(s3) and {dot over(X)}*_(s4) are the first derivatives of the virtual controllers X*_(s3)and X*_(s4), respectively, T_(m)(t) and T_(s)(t) are forward time delay(from the master robot to the slave robot) and backward time delay (fromthe slave robot to the master robot), respectively. α_(m) and α_(s) aredamping coefficients which are positive constants. k_(m), k_(s)>0 areproportional coefficients. S_(s) ⁻¹ and S_(s) ^(T) are the inversematrix and the transpose matrix of a diagonal positive-definite constantmatrix S_(s) which contains the joint stiffness of the slave robot,respectively. J_(s) is the diagonal constant matrix of the moments ofactuator inertia. k₁ and k₂ are selected to be positive constants.

Step 3: designing high-dimensional uniform accurate differentiators tocarry out a precise difference to the first virtual controller and thesecond virtual controller.

Letting X₁=X*_(s3), X₂={dot over (X)}*_(s3), σ₁=X₁−Y₁, σ₂=X₂−Y₂,

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{1} = X_{2}} \\{{\overset{.}{X}}_{2} = {\overset{¨}{X}}_{s\; 3}^{*}}\end{matrix} \right. & (16) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{1} = {{\lambda_{1}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} + {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + Y_{2}}} \\{{\overset{.}{Y}}_{2} = {\alpha_{1}\frac{\sigma_{1}}{\sigma_{1}}}}\end{matrix} \right. & (17)\end{matrix}$

with Equations (16) and (17),

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{1} = {{{- \lambda_{1}}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} - {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + \sigma_{2}}} \\{{\overset{.}{\sigma}}_{2} = {{{- \alpha_{1}}\frac{\sigma_{1}}{\sigma_{1}}} + {\overset{¨}{X}}_{s\; 3}^{*}}}\end{matrix} \right. & (18)\end{matrix}$

is derived, where X*_(s3) denotes the first virtual controller, {dotover (X)}*_(s3) denotes the first derivative of the first virtualcontroller X*_(s3), {umlaut over (X)}_(s3) denotes the second derivativeof the first virtual controller X*_(s3), Y₁ is an estimate of the firstvirtual controller X*_(s3), Y₂ is an estimate of {dot over (X)}*_(s3),σ₁ and σ₂ are estimation errors, λ₁, λ₂, α₁>0 are system control gains,P>1 is a constant, {umlaut over (X)}*_(s3) is supposed to be bounded andsatisfies ∥{umlaut over (X)}*_(s3)∥≤L₃ with a known constant L₃>0. Ifthe parameters are selected to satisfy the conditions α₁>4L₃,λ₁>√{square root over (2α₁)}, the estimation errors σ₁, σ₂ will convergeto the origin quickly, thus a precise difference value {dot over(X)}*_(s3) of X*_(s3) is obtained.

Letting X₃=X*_(s4), X₄={dot over (X)}*_(s4), σ₃=X₃−Y₃, σ₄=X₄−Y₄,

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{3} = X_{4}} \\{{\overset{.}{X}}_{4} = {\overset{¨}{X}}_{s\; 4}^{*}}\end{matrix} \right. & (19) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{3} = {{\lambda_{3}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} + {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + Y_{4}}} \\{{\overset{.}{Y}}_{4} = {\alpha_{2}\frac{\sigma_{3}}{\sigma_{3}}}}\end{matrix} \right. & (20)\end{matrix}$

with Equations (19) and (20),

$\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{3} = {{{- \lambda_{3}}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} - {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + \sigma_{4}}} \\{{\overset{.}{\sigma}}_{4} = {{{- \alpha_{2}}\frac{\sigma_{3}}{\sigma_{3}}} + {\overset{¨}{X}}_{s\; 4}^{*}}}\end{matrix} \right. & (21)\end{matrix}$

is derived, where X*_(s4) denotes the second virtual controller, {dotover (X)}*_(s4) denotes the first derivative of the second virtualcontroller X*_(s4), {umlaut over (X)}*_(s4) denotes the secondderivative of the second virtual controller X*_(s4), Y₃ is an estimateof the second virtual controller X*_(s4), Y₄ is an estimate of {dot over(X)}*_(s4), σ₃ and σ₄ are estimation errors, λ₃, λ₄, α₂>0 are systemcontrol gains, {umlaut over (X)}*_(s4) is supposed to be bounded andsatisfies ∥{umlaut over (X)}*_(s4)∥≤L₄ with a known positive constantL₄>0. If the parameters are selected to satisfy the conditions α₂>4L₄,λ₃>√{square root over (2α₂)}, the estimation errors σ₃, σ₄ will convergeto the origin quickly, thus a precise difference value {dot over(X)}*_(s4) of X*_(s4) is obtained.

Step 4: Establishing the delay-dependent system stability criteria byconstructing Lyapunov Equations, providing the criteria for selectingcontroller parameters, and realizing the global stability of themaster-slave robot system with flexible joints and time-varying delays.

The Lyapunov Equation is selected as

$\begin{matrix}{{V_{11} = {{X_{m2}^{T}{M_{m}\left( X_{m1} \right)}X_{m2}} + {\frac{k_{m}}{k_{s}}X_{s2}^{T}{M_{s}\left( X_{s1} \right)}X_{s2}} + {2\left( {{U_{m}\left( X_{m1} \right)} - \beta_{m}} \right)} + {\frac{2k_{m}}{k_{s}}\left( {{U_{s}\left( X_{s1} \right)} - \beta_{s}} \right)} + {2{\int_{0}^{t}{\left( {{{- {X_{m2}^{T}(\sigma)}}{F_{h}(\sigma)}} + {\frac{k_{m}}{k_{s}}{X_{s2}^{T}(\sigma)}{F_{e}(\sigma)}}} \right)d\;\sigma}}}}}\mspace{79mu}{V_{12} = {{k_{m}\left( {X_{m1} - X_{s1}} \right)}^{T}\left( {X_{m1} - X_{s1}} \right)}}{V_{13} = {{\int_{- {\overset{\_}{T}}_{m}}^{0}{\int_{t + \theta}^{t}{{X_{m2}^{T}(\xi)}Z{X_{m2}(\xi)}d\;\xi\; d\;\theta}}} + {\int_{- {\overset{\_}{T}}_{s}}^{0}{\int_{t + \theta}^{t}{{X_{s2}^{T}(\xi)}P{X_{s2}(\xi)}d{\xi d\theta}}}}}}\mspace{79mu}{V_{1} = {V_{11} + V_{12} + V_{13}}}\mspace{79mu}{V_{2} = {V_{1} + {\frac{1}{2}\left( {X_{s3} - X_{s\; 3}^{*}} \right)^{T}\left( {X_{s3} - X_{s3}^{*}} \right)}}}\mspace{79mu}{V_{3} = {V_{2} + {\frac{1}{2}\left( {X_{s4} - X_{s4}^{*}} \right)^{T}\left( {X_{s4} - X_{s4}^{*}} \right)}}}} & (22)\end{matrix}$

The time derivative of V₃ is given by

$\begin{matrix}{{\overset{.}{V}}_{3} \leq {{{X_{m2}^{T}\left( {{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m2}} + {{X_{s2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s2}}\mspace{79mu} - {{k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}^{T}\left( {X_{s3} - X_{s3}^{*}} \right)} - {{k_{2}\left( {X_{s4} - X_{s4}^{*}} \right)}^{T}\left( {X_{s4} - X_{s4}^{*}} \right)}}} & (23)\end{matrix}$

When the controller parameters α_(m), α_(s), k_(m), k_(s), I, T _(m), T_(s), Z, P are selected such that the following conditions hold,

${{{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$

the joint and motor velocities {dot over (q)}_(m), {dot over (q)}_(s),{dot over (θ)}_(s) and position error q_(m)−q_(s) of the master-slaverobot system with flexible joints are all bounded.

If the force F_(h) exerted by an operator to the master robot and aforce F_(e) exerted by the external environment to the slave robot areboth zero, the controllers are designed as follows:

$\begin{matrix}{{{\tau_{m} = {{- {k_{m}\left( {X_{m1} - {X_{s1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {\alpha_{m}X_{m2}} + {G_{m}\left( X_{m1} \right)}}}{\tau_{s} = {{S_{s}\left( {X_{s3} - X_{s1}} \right)} + {J_{s}\left( {{\overset{.}{X}}_{s4}^{*} - \left( {X_{s3} - X_{s3}^{*}} \right) - {k_{2}\left( {X_{s4} - X_{s4}^{*}} \right)}} \right)}}}}{X_{s3}^{*} = {X_{s1} + {S_{s}^{- 1}\left( {{k_{s}\left( {{X_{m1}\left( {t - {T_{m}(t)}} \right)} - X_{s1}} \right)} - {\alpha_{s}X_{s2}} + {G_{s}\left( X_{s1} \right)}} \right)}}}{X_{s4}^{*} = {{\overset{.}{X}}_{s3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}}}} & (24)\end{matrix}$

where X*_(s3) and X*_(s4) are the first controller and the secondvirtual controller, respectively, {dot over (X)}*_(s3) and {dot over(X)}*_(s4) are the first derivatives of the virtual controllers X*_(s3)and X*_(s4), respectively, T_(m)(t) and T_(s)(t) are forward time delay(from the master robot to the slave robot) and backward time delay (fromthe slave robot to the master robot), respectively, α_(m) and α_(s) aredamping coefficients which are positive constants, k_(m), k_(s)>0 areproportional coefficients, S_(s) ⁻¹ and S_(s) ^(T) are the inversematrix and the transpose matrix of a diagonal positive-definite constantmatrix S_(s) which contains the joint stiffness of the slave robot,respectively, G_(m)(X_(m1)), G_(s)(X_(s1)) are the gravity torques ofthe master robot and the slave robot, J_(s) is a diagonal constantmatrix of the moment of actuator inertia, and k₁ and k₂ are selected tobe positive constants.

When the controller parameters α_(m), α_(s), k_(m), k_(s), I, T _(m), T_(s), Z, P are selected such that the following conditions hold,

${{{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$

it can be guaranteed that the joint and motor velocities {dot over(q)}_(m), {dot over (q)}_(s), {dot over (θ)}_(s) and the position errorq_(m)−q_(s) of the master-slave robot system with flexible joints willconverge to zero asymptotically and that the global master-slave robotsystem with flexible joints will asymptotically become stable.

For a master-slave robot system with flexible joints, first a controlcommand is sent by the operator to the master robot, and the masterrobot is controlled by the proportional damping controller τ_(m) andtransmits the received control command to the slave robot through thenetwork information transmission channel. Then the slave robotcontrolled by the full-state feedback controller τ_(s) executes thecontrol command on the external environment, feedbacks the measuredposition and force information to the operator in time, thus aclosed-loop system is formed and tasks can be completed effectively.

Based on the backstepping technique and the high-dimension uniformaccurate differentiators, a full-state feedback controller is proposedfor a flexible master-slave robot system. With this flexiblemaster-slave robot system, accurate position tracking performance isachieved in the global scope. Additionally, the global asymptoticconvergence is guaranteed and the robustness of the closed-loopmaster-slave system is improved.

The above-mentioned embodiment merely describes the preferred embodimentof the present invention. The scope of the invention should not belimited by the described embodiment. It is intended that various changesand modifications to the technical solutions of the present inventionmade by those skilled in the art without departing from the spirit ofthe present invention shall fall within the protection scope determinedby the claims of the present invention.

1. A full-state control method for a master-slave robot system withflexible joints and time-varying delays, wherein the method comprising:connecting a master robot and a slave robot through the network to forma teleoperation system; measuring system parameters of the master robotand the slave robot, among which the position and velocity informationof joints and motors are measured in real-time; designing a firstvirtual controller and a second virtual controller, respectively,wherein the first virtual controller is X*_(s3)=X_(s1)+S_(s)⁻¹(k_(s)(X_(m1)(t−T_(m)(t))−X_(s1))−α_(s)X_(s2)), the second virtualcontroller is${X_{s4}^{*} = {{\overset{.}{X}}_{s3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}}},$the subscripts m and s denotes the master robot and the slave robot,respectively, X*_(s3) and X*_(s4) are virtual controllers, {dot over(X)}*_(s3) is the first derivative of the virtual controller X*_(s3),X_(m1) is a vector of joint displacements of the master robot, X_(s1) isa vector of joint displacements of the slave robot, X_(s2) is a vectorof joint velocities of the slave robot, X_(s3) is a vector of motorpositions of the slave robot, T_(m)(t) and T_(s)(t) denote forward timedelay from the master robot to slave robot and feedback time delay fromthe slave robot to the master robot, respectively, α_(s) is a dampingcoefficient which is a positive constant, k_(m), k_(s)>0 areproportional coefficients, S_(s) ⁻¹ and S_(s) ^(T) are the inversematrix and the transpose matrix of a diagonal positive-definite constantmatrix S_(s) which contains the joint stiffness of the slave robot,respectively, and k₁ is selected to be a positive constant; designinghigh-dimensional uniform accurate differentiators to carry out a precisedifference to the first virtual controller and the second virtualcontroller, wherein letting X₁=X*_(s3), X₂={dot over (X)}*_(s3),σ₁=X₁−Y₁, σ₂=X₂−Y₂, $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{1} = X_{2}} \\{{\overset{.}{X}}_{2} = {\overset{¨}{X}}_{s\; 3}^{*}}\end{matrix} \right. & (1) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{1} = {{\lambda_{1}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} + {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + Y_{2}}} \\{{\overset{.}{Y}}_{2} = {\alpha_{1}\frac{\sigma_{1}}{\sigma_{1}}}}\end{matrix} \right. & (2)\end{matrix}$ with (1) and (2), $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{1} = {{{- \lambda_{1}}\frac{\sigma_{1}}{{\sigma_{1}}^{1/2}}} - {\lambda_{2}\sigma_{1}{\sigma_{1}}^{P - 1}} + \sigma_{2}}} \\{{\overset{.}{\sigma}}_{2} = {{{- \alpha_{1}}\frac{\sigma_{1}}{\sigma_{1}}} + {\overset{¨}{X}}_{s\; 3}^{*}}}\end{matrix} \right. & (3)\end{matrix}$ is derived, where X*_(s3) denotes the virtual controller,{dot over (X)}*_(s3) denotes the first derivative of the virtualcontroller X*_(s3), {umlaut over (X)}*_(s3) denotes the secondderivative of the virtual controller X*_(s3), Y₁ is an estimate of thevirtual controller X*_(s3), Y₂ is an estimate of {dot over (X)}*_(s3),σ₁ and σ₂ are estimation errors, λ₁, λ₂, α₁>0 are system control gains,P>1 is a constant, {umlaut over (X)}*_(s3) is supposed to be bounded andsatisfies ∥{umlaut over (X)}*_(s3)∥≤L₃ with a known constant L₃>0, andif the parameters are selected to satisfy the conditions α₁>4L₃,λ₁>√{square root over (2α₁)}, the estimation errors σ₁, σ₂ will globallyconverge to the origin quickly, thus a precise difference value {dotover (X)}*_(s3) of X*_(s3) is obtained, wherein letting X₃=X*_(s4),X₄={dot over (X)}*_(s4), σ₃=X₃−Y₃, σ₄=X₄−Y₄, $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{X}}_{3} = X_{4}} \\{{\overset{.}{X}}_{4} = {\overset{¨}{X}}_{s\; 4}^{*}}\end{matrix} \right. & (4) \\\left\{ \begin{matrix}{{\overset{.}{Y}}_{3} = {{\lambda_{3}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} + {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + Y_{4}}} \\{{\overset{.}{Y}}_{4} = {\alpha_{2}\frac{\sigma_{3}}{\sigma_{3}}}}\end{matrix} \right. & (5)\end{matrix}$ with (4) and (5), $\begin{matrix}\left\{ \begin{matrix}{{\overset{.}{\sigma}}_{3} = {{{- \lambda_{3}}\frac{\sigma_{3}}{{\sigma_{3}}^{1/2}}} - {\lambda_{4}\sigma_{3}{\sigma_{3}}^{P - 1}} + \sigma_{4}}} \\{{\overset{.}{\sigma}}_{4} = {{{- \alpha_{2}}\frac{\sigma_{3}}{\sigma_{3}}} + {\overset{¨}{X}}_{s\; 4}^{*}}}\end{matrix} \right. & (6)\end{matrix}$ is derived, where X*_(s4) denotes a virtual controller,{dot over (X)}*_(s4) is the first derivative of the virtual controllerX*_(s4), {umlaut over (X)}*_(s4) is the second derivative of the virtualcontroller X*_(s4), Y₃ is an estimate of the virtual controller X*_(s4),Y₄ is an estimate of {dot over (X)}*_(s4), σ₃ and σ₄ are estimationerrors, λ₃, λ₄, α₂>0 are system control gains, {umlaut over (X)}*_(s4)is supposed to be bounded and satisfies ∥{umlaut over (X)}*_(s4)∥≥L₄with a known positive constant L₄>0, if the parameters are selected tosatisfy the conditions α₂>4L₄, λ₃>√{square root over (2α₂)}, theestimation errors σ₃, σ₄ will globally converge to the origin quickly,thus a precise difference value {dot over (X)}*_(s4) of X*_(s4) isobtained; designing controllers of the master-slave robot system withflexible joints by applying the backstepping method,τ_(m) = −k_(m)(X_(m1) − X_(s1)(t − T_(s)(t))) − α_(m)X_(m2)$\tau_{s} = {{S_{s}\left( {X_{s3} - X_{s1}} \right)} + {J_{s}\left( {{\overset{.}{X}}_{s4}^{*} - \left( {X_{s3} - X_{s3}^{*}} \right) - {k_{2}\left( {X_{s4} - X_{s4}^{*}} \right)}} \right)}}$X_(s3)^(*) = X_(s1) + S_(s)⁻¹(k_(s)(X_(m1)(t − T_(m)(t)) − X_(s1)) − α_(s)X_(s2))$X_{s4}^{*} = {{\overset{.}{X}}_{s3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s2}} - {k_{1}\left( {X_{s3} - X_{s3}^{*}} \right)}}$where τ_(m) and τ_(s) are control torques provided by the controllers,X*_(s3) and X*_(s4) are virtual controllers, {dot over (X)}*_(s3) and{dot over (X)}*_(s4) are the first derivative of the virtual controllersX*_(s3) and X*_(s4), respectively, Z_(m1) is a vector of jointdisplacement of the master robot, X_(m2) is a vector of joint velocitiesof the master robot, X_(s1) is a vector of joint displacement of theslave robot, X_(s2) is a vector of joint velocities of the slave robot,X_(s3) is a vector of motor positions of the slave robot, X_(s4) is avector of motor velocities of the slave robot, T_(m)(t) and T_(s)(t) areforward time delay from the master robot to the slave robot and feedbacktime delay from the slave robot to the master robot, respectively, α_(m)and α_(s) are damping coefficients which are positive constants, k_(m),k_(s)>0 are proportional coefficients, S_(s) ⁻¹ and S_(s) ^(T) are theinverse matrix and the transpose matrix of a diagonal positive-definiteconstant matrix S_(s) which contains the joint stiffness of the slaverobot, respectively, J_(s) is the diagonal constant matrix of the momentof actuator interia, and k₁ and k₂ are selected to be positiveconstants; and establishing the delay-dependent system stabilitycriteria by constructing Lyapunov Equations, and realizing the globalstability of the master-slave robot system with flexible joints andtime-varying delays, wherein when the controller parameters are selectedsuch that the following inequalities hold,${{{{- 2}\;\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$the joint and motor velocities {dot over (q)}_(m), {dot over (q)}_(s),{dot over (θ)}_(s) and the position error q_(m)−q_(s) of themaster-slave robot system with flexible joints are all bounded, when aforce F_(h) exerted by an operator to the master robot and a force F_(e)exerted by the external environment to the slave robot are both zero,the controllers are designed as follows: $\begin{matrix}{{{\tau_{m} = {{- {k_{m}\left( {X_{m\; 1} - {X_{s\; 1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {\alpha_{m}X_{m\; 2}} + {G_{m}\left( X_{m\; 1} \right)}}}\tau_{s} = {{S_{s}\left( {X_{s\; 3} - X_{s\; 1}} \right)} + {J_{s}\left( {{\overset{.}{X}}_{s\; 4}^{*} - \left( {X_{s\; 3} - X_{s\; 3}^{*}} \right) - {k_{2}\left( {X_{s\; 4} - X_{s\; 4}^{*}} \right)}} \right)}}}{X_{s\; 3}^{*} = {X_{s\; 1} + {S_{s}^{- 1}\left( {{k_{s}\left( {{X_{m\; 1}\left( {t - {T_{m}(t)}} \right)} - X_{s\; 1}} \right)} - {\alpha_{s}X_{s\; 2}} + {G_{s}\left( X_{s\; 1} \right)}} \right)}}}{X_{s\; 4}^{*} = {{\overset{.}{X}}_{s\; 3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s\; 2}} - {k_{1}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}}}} & (7)\end{matrix}$ where τ_(m) and τ_(s) are control torques provided by thecontrollers, X*_(s3) and X*_(s4) are virtual controllers, {dot over(X)}*_(s3) and {dot over (X)}*₄ are the first derivatives of the virtualcontrollers X*_(s3) and X*_(s4), respectively, T_(m)(t) and T_(s)(t) areforward time delay from the master robot to the slave robot and feedbacktime delay from the slave robot to the master robot, respectively, α_(m)and α_(s) are damping coefficients which are positive constants, k_(m),k_(s)>0 are proportional coefficients, S_(s) ⁻¹ and S_(s) ^(T) are theinverse matrix and the transpose matrix of a diagonal positive-definiteconstant matrix S_(s) which contains the joint stiffness of the slaverobot, respectively, G_(m)(X_(m1)), G_(s)(X_(s1)) are the gravitationaltorques of the master robot and the slave robot, J_(s) is a diagonalconstant matrix of the moment of actuator inertia, and k₁ and k₂ areselected to be positive constants, when the controller parameters areselected thus the following inequalities hold,${{{{- 2}\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$it can be guaranteed that the joint and motor velocities {dot over(q)}_(m), {dot over (q)}_(s), {dot over (θ)}_(s) and the position errorq_(m)−q_(s) of the master-slave robot system with flexible joints willconverge to zero asymptotically and that the global master-slave robotsystem with flexible joints will asymptotically become stable, where Iis the identity matrix, Z⁻¹ and P⁻¹ are the inverse matrices of positivedefinite matrices Z and P, respectively, α_(m) and α_(s) are dampingcoefficients which are positive constants, k_(m), k_(s)>0 areproportional coefficients, and it is supposed that the time delayT_(m)(t) and T_(s)(t) are bounded, i.e. there are positive scalars T_(m) and T _(s), such that the inequalities T_(m)(t)≤T _(m), T_(s)(t)≤T_(s) hold.
 2. The full-state control method of the master-slave robotsystem with flexible joints and time-varying delays of claim 1, whereinthe system parameters of the master robot and the slave robot includethe length and the mass of the manipulators of the master robot and theslave robot, the positions and the velocities of the robot joints andmotor of the master robot and the slave robot, and the forces exerted bythe operator and the external environment measured by using forcesensors.
 3. The full-state control method of the master-slave robotsystem with flexible joints and time-varying delays of claim 1, whereinthe design process of the first virtual controller and the secondvirtual controller is as follows: as for the first virtual controller,the first Lyapunov Equation is selected as follows, $\begin{matrix}{\mspace{79mu}{{V_{1} = {V_{11} + V_{12} + V_{13}}}{V_{11} = {{X_{m\; 2}^{T}{M_{m}\left( X_{m\; 1} \right)}X_{m\; 2}} + {\frac{k_{m}}{k_{s}}X_{s\; 2}^{T}{M_{s}\left( X_{s\; 1} \right)}X_{s\; 2}} + {2\left( {{U_{m}\left( X_{m\; 1} \right)} - \beta_{m}} \right)} + {\frac{2\; k_{m}}{k_{s}}\left( {{U_{s}\left( X_{s\; 1} \right)} - \beta_{s}} \right)} + {2{\int_{0}^{t}{\left( {{{- {X_{m\; 2}^{T}(\sigma)}}{F_{h}(\sigma)}} + {\frac{k_{m}}{k_{s}}{X_{s\; 2}^{T}(\sigma)}{F_{e}(\sigma)}}} \right)d\;\sigma}}}}}\mspace{79mu}{V_{12} = {{k_{m}\left( {X_{m\; 1} - X_{s\; 1}} \right)}^{T}\left( {X_{m\; 1} - X_{s\; 1}} \right)}}{V_{13} = {{\int_{- {\overset{\_}{T}}_{m}}^{0}{\int_{t + \theta}^{t}{{X_{m\; 2}^{T}(\xi)}{{ZX}_{m\; 2}(\xi)}d\;\xi\; d\;\theta}}} + {\int_{- {\overset{\_}{T}}_{s}}^{0}{\int_{t + \theta}^{t}{{X_{s\; 2}^{T}(\xi)}{{PX}_{s\; 2}(\xi)}d\;\xi\; d\;\theta}}}}}}} & (8)\end{matrix}$ where the integral terms satisfy ∫₀ ^(t)−X_(m2)^(T)(σ)F_(h)(σ)dσ≥0, ∫₀ ^(t)X_(s2) ^(T)(σ)F_(e)(σ)dσ≥0, Z and P arepositive definite matrices, it is supposed that the time delay T_(m)(t)and T_(s)(t) are bounded, i.e. there are positive scalars T _(m) and T_(s), such that T_(m)(t)≤T _(m), T_(s)(t)≤T _(s), M_(m)(X_(m1)) andM_(s)(X_(s1)) are positive-definite inertia matrices of the master robotand the slave robot, respectively, U_(m)(X_(m1)) and U_(s)(X_(s1)) arethe potential energy of the master robot and the slave robot satisfying${{G_{m}\left( X_{m\; 1} \right)} = \frac{\partial{U_{m}\left( X_{m\; 1} \right)}}{\partial X_{m\; 1}}},{{G_{s}\left( X_{s\; 1} \right)} = \frac{\partial{U_{s}\left( X_{s\; 1} \right)}}{\partial X_{s\; 1}}},$there are positive scalars β_(m) and β_(s) such thatU_(m)(X_(m1))≥β_(m), U_(s)(X_(s1))≥β_(s), and k_(m), k_(s)>0 areproportional coefficients, a time derivative of V₁₁ is given by$\begin{matrix}{{\overset{.}{V}}_{11} = {{{{- 2}\; k_{m}{X_{m\; 2}^{T}\left( {X_{m\; 1} - {X_{s\; 1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {2\;\alpha_{m}X_{m\; 2}^{T}X_{m\; 2}} + {2\frac{k_{m}}{k_{s}}X_{s\; 2}^{T}{S_{s}\left( {X_{s\; 3} - X_{s\; 1}} \right)}}} = {{{- 2}\; k_{m}{X_{m\; 2}^{T}\left( {X_{m\; 1} - {X_{s\; 1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {2\;\alpha_{m}X_{m\; 2}^{T}X_{m\; 2}} + {2\frac{k_{m}}{k_{s}}{X_{s\; 2}^{T}\left( {{S_{s}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)} + {S_{s}X_{s\; 3}^{*}} - {S_{s}X_{s\; 1}}} \right)}}}}} & (9)\end{matrix}$ where S_(s) is a diagonal positive-definite constantmatrix that contains the joint stiffness of the slave robot, S_(s) ⁻¹ isthe inverse matrix of S and the first virtual controllerX* _(s3) =X _(s1) +S _(s) ⁻¹(k _(s)(X _(m1)(t−T _(m)(t))−X _(s1))−α_(s)X _(s2)) is derived from equation (9), as for the second virtualcontroller, the second Lyapunov Equation is selected as follows,V ₂ =V ₁+½(X _(s3) −X* _(s3))^(T)(X _(s3) −X* _(s3))  (10) The timederivative of V₂ is given by{dot over (V)} ₂ ={dot over (V)} ₁+(X _(s3) −X* _(s3))^(T)(X _(s4) −{dotover (X)}* _(s3))={dot over (V)} ₁+(X _(s3) −X* _(s3))^(T)(X _(s4) −X*_(s4) +X* _(s4) −{dot over (X)}* _(s3))  (11) and the second virtualcontroller$X_{s\; 4}^{*} = {{\overset{.}{X}}_{s\; 3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s\; 2}} - {k_{1}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}}$is derived from equation (11).
 4. The full-state control method of themaster-slave robot system with flexible joints and time-varying delaysof claim 1, wherein a delay-dependent system stability criteria isestablished by constructing the Lyapunov Equations, thus a criteria forselecting controller parameters are provided, the detailed steps are asfollows, selecting the first Lyapunov Equation as follows:$\begin{matrix}{\mspace{76mu}{{V_{1} = {V_{11} + V_{12} + V_{13}}}{V_{11} = {{X_{m\; 2}^{T}{M_{m}\left( X_{m\; 1} \right)}X_{m\; 2}} + {\frac{k_{m}}{k_{s}}X_{s\; 2}^{T}{M_{s}\left( X_{s\; 1} \right)}X_{s\; 2}} + {2\left( {{U_{m}\left( X_{m\; 1} \right)} - \beta_{m}} \right)} + {\frac{2\; k_{m}}{k_{s}}\left( {{U_{s}\left( X_{s\; 1} \right)} - \beta_{s}} \right)} + {2{\int_{0}^{t}{\left( {{{- {X_{m\; 2}^{T}(\sigma)}}{F_{h}(\sigma)}} + {\frac{k_{m}}{k_{s}}{X_{s\; 2}^{T}(\sigma)}{F_{e}(\sigma)}}} \right)d\;\sigma}}}}}\mspace{79mu}{V_{12} = {{k_{m}\left( {X_{m\; 1} - X_{s\; 1}} \right)}^{T}\left( {X_{m\; 1} - X_{s\; 1}} \right)}}{V_{13} = {{\int_{- {\overset{\_}{T}}_{m}}^{0}{\int_{t + \theta}^{t}{{X_{m\; 2}^{T}(\xi)}{{ZX}_{m\; 2}(\xi)}d\;\xi\; d\;\theta}}} + {\int_{- {\overset{\_}{T}}_{s}}^{0}{\int_{t + \theta}^{t}{{X_{s\; 2}^{T}(\xi)}{{PX}_{s\; 2}(\xi)}d\;\xi\; d\;\theta}}}}}}} & (18)\end{matrix}$ where the integral terms satisfy ∫₀ ^(t)−X_(m2)^(T)(σ)F_(h)(σ)dσ≥0, ∫₀ ^(t)X_(s2) ^(T)(σ)F_(e)(σ)dσ≥0, Z and P arepositive definite matrices, it is supposed that the time delay T_(m)(t)and T_(s)(t) are bounded, i.e. there are positive scalars T _(m) and T_(s), such that T_(m)(t)≤T _(m), T_(s)(t)≤T _(s), M_(m)(X_(m1)) andM_(s)(X_(s1)) are the positive-definite inertia matrices of the masterrobot and the slave robot, respectively, U_(m)(X_(m1)) and U_(s)(X_(s1))are the potential energy of the master robot and the slave robotsatisfying${{G_{m}\left( X_{m\; 1} \right)} = \frac{\partial{U_{m}\left( X_{m\; 1} \right)}}{\partial X_{m\; 1}}},{{G_{s}\left( X_{s\; 1} \right)} = \frac{\partial{U_{s}\left( X_{s\; 1} \right)}}{\partial X_{s\; 1}}},$there are positive scalars β_(m) and β_(s) such thatU_(m)(X_(m1))≥β_(m), U_(s)(X_(s1))≥β_(s), and k_(m), k_(s)>0 areproportional coefficients, the time derivative of V₁ is given by$\begin{matrix}{{\overset{.}{V}}_{1} = {{{\overset{.}{V}}_{11} + {\overset{.}{V}}_{12} + {\overset{.}{V}}_{13}} \leq {{{X_{m\; 2}^{T}\left( {{{- 2}\;\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m\; 2}} + {{X_{s\; 2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s\; 2}} + {2\frac{k_{m}}{k_{s}}X_{s\; 2}^{T}{S_{s}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}}}}} & (19)\end{matrix}$ where I is the identity matrix, Z⁻¹ and P⁻¹ are theinverse matrices of positive definite matrices Z and P, respectively,S_(s) is a diagonal positive-definite constant matrix which contains thejoint stiffness of the slave robot, α_(m) and α_(s) are dampingcoefficients which are positive constants, k_(m), k_(s)>0 areproportional coefficients, it is supposed that the time delay T_(m)(t)and T_(s)(t) are bounded, i.e. there are positive scalars T _(m) and T_(s), such that T_(m)(t)≤T _(m), T_(s)(t)≤T _(s), and X*_(s3) is thefirst virtual controller; selecting the second Lyapunov Equation asfollows:V ₂ =V ₁+½(X _(s3) −X* _(s3))^(T)(X _(s3) −X* _(s3))  (20) the timederivative of V₂ is given by $\begin{matrix}{{{\overset{.}{V}}_{2} \leq {{{X_{m\; 2}^{T}\left( {{{- 2}\;\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m\; 2}} + {{X_{s\; 2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s\; 2}} + {\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)^{T}\left( {X_{s\; 4} - X_{s\; 4}^{*}} \right)} - {{k_{1}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}^{T}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}}};} & (21)\end{matrix}$ selecting the third Lyapunov Equation as follows:V ₃ =V ₂+½(X _(s4) −X* _(s4))^(T)(X _(s4) −X* _(s4))  (22) the timederivative of V₃ is $\begin{matrix}{{{\overset{.}{V}}_{3} \leq {{{X_{m\; 2}^{T}\left( {{{- 2}\;\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} \right)}X_{m\; 2}} + {{X_{s\; 2}^{T}\left( {{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} \right)}X_{s\; 2}} - {{k_{1}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}^{T}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)} - {{k_{2}\left( {X_{s\; 4} - X_{s\; 4}^{*}} \right)}^{T}\left( {X_{s\; 4} - X_{s\; 4}^{*}} \right)}}},} & (23)\end{matrix}$ when the controller parameters are selected such that thefollowing inequalities hold,${{{{- 2}\;\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$the joint and motor velocities {dot over (q)}_(m), {dot over (q)}_(s),{dot over (θ)}_(s) and the position error q_(m)−q_(s) of themaster-slave robot system with flexible joints are all bounded, when aforce F_(h) exerted by an operator to the master robot and a force F_(e)exerted by the external environment are both zero, the controllers aredesigned as follows: $\begin{matrix}{{{\tau_{m} = {{- {k_{m}\left( {X_{m\; 1} - {X_{s\; 1}\left( {t - {T_{s}(t)}} \right)}} \right)}} - {\alpha_{m}X_{m\; 2}} + {G_{m}\left( X_{m\; 1} \right)}}}\tau_{s} = {{S_{s}\left( {X_{s\; 3} - X_{s\; 1}} \right)} + {J_{s}\left( {{\overset{.}{X}}_{s\; 4}^{*} - \left( {X_{s\; 3} - X_{s\; 3}^{*}} \right) - {k_{2}\left( {X_{s\; 4} - X_{s\; 4}^{*}} \right)}} \right)}}}{X_{s\; 3}^{*} = {X_{s\; 1} + {S_{s}^{- 1}\left( {{k_{s}\left( {{X_{m\; 1}\left( {t - {T_{m}(t)}} \right)} - X_{s\; 1}} \right)} - {\alpha_{s}X_{s\; 2}} + {G_{s}\left( X_{s\; 1} \right)}} \right)}}}{X_{s\; 4}^{*} = {{\overset{.}{X}}_{s\; 3}^{*} - {2\frac{k_{m}}{k_{s}}S_{s}^{T}X_{s\; 2}} - {k_{1}\left( {X_{s\; 3} - X_{s\; 3}^{*}} \right)}}}} & (24)\end{matrix}$ where τ_(m) and τ_(s) are control torques provided by thecontrollers, X*_(s3) and X*_(s4) are virtual controllers, {dot over(X)}*_(s3) and {dot over (X)}*_(s4) are the first derivatives of thevirtual controllers X*_(s3) and X*_(s4), respectively, T_(m)(t) andT_(s)(t) are forward time delay from the master robot to the slave robotand feedback time delay from the slave robot to the master robot,respectively, α_(n), and α_(s) are damping coefficients which arepositive constants, k_(m), k_(s)>0 are proportional coefficients, S_(s)⁻¹ and S_(s) ^(T) are the inverse matrix and the transpose matrix of adiagonal positive-definite constant matrix S_(s) which contains thejoint stiffness of the slave robot, respectively, G_(m)(X_(m1)),G_(s)(X_(s1)) are the gravity matrices of the master robot and the slaverobot, respectively, J_(s) is the diagonal constant matrix of themoments of actuator inertia, G_(m)(X_(m1)), G_(s)(X_(s1)) are thegravity torques of the master robot and the slave robot, respectively,and k₁ and k₂ are selected to be positive constants, when the controllerparameters are selected such that the following inequalities hold,${{{{- 2}\;\alpha_{m}I} + {{\overset{\_}{T}}_{m}Z} + {{\overset{\_}{T}}_{s}k_{m}^{2}P^{- 1}}} < 0},{{{{- 2}\frac{k_{m}\alpha_{s}}{k_{s}}I} + {{\overset{\_}{T}}_{s}P} + {{\overset{\_}{T}}_{m}k_{m}^{2}Z^{- 1}}} < 0}$it is guaranteed that the joint and motor velocities {dot over (q)}_(m),{dot over (q)}_(s), {dot over (θ)}_(s) and the position errorq_(m)−q_(s) of the master-slave robot system with flexible joints willconverge to zero asymptotically and that the global master-slave robotsystem with flexible joints will asymptotically become stable.